Bandwidth is the amount of data carried between two points, like Michael said, but it can be measured in two different streams: up and down.

Upstream, aka upload, is when you are sending data from your machine to a remote location.  Downstream, aka download, is where the remote location is sending data to your machine.

In aspects of hosting, a lot of providers who offer “cloud” or VPS services will specify if they count local bandwidth against your allotment too.  Which, usually they don’t as the bandwidth used is like passing an apple from your right hand to the left, where as non-local bandwidth is like passing an apple from one end of a building to another.